ch13 不确定性 人工智能课程 北大计算机研究所

Axioms of probability


For any propositions A, B
0 ≤ P(A) ≤ 1 P(true) = 1 and P(false) = 0 P(A B) = P(A) + P(B) - P(A B)
Axioms of probability (traditional)

For any proposition φ, sum the atomic events where it is true: P(φ) = Σω:ω╞φ P(ω)

Inference by enumeration

Start with the joint probability distribution:
Methods for handling uncertainty (contd)

Probability

Model agent's degree of belief Ex: Given the available evidence, A25 will get me there on time with probability 0.04
―A25 will get me there on time if there's no accident on the bridge and it doesn't rain and my tires remain intact etc etc.‖ (A1440 might reasonably be said to get me there on time but I'd have to stay overnight in the
A25 |→0.3 get there on time Sprinkler |→ 0.99 WetGrass WetGrass |→ 0.7 Rain

Issues: Problems with combination, e.g., Sprinkler causes Rain??
Dempster-Shafer theory

Issues: What assumptions are reasonable? How to handle contradiction?
Assume A25 works unless contradicted by evidence

Rules with fudge factors


Probability
Probabilistic assertions summarize effects of

laziness: failure to enumerate exceptions, qualifications, etc. ignorance: lack of relevant facts, initial conditions, etc.

概率



概率公理 使用全概率分布进行推理 独立性 贝叶斯法则及其应用
Methods for handling uncertainty

Default or nonmonotonic logic (Section 10.7)

Assume my car does not have a flat tire
客观主义者对概率的看法

概率是宇宙的真实方面：它是物体的行 为表现为特定方式的倾向，而不仅仅是 对观察者信心的描述在许多情景下不可


频率主义者的观点：断言“有牙洞的概率是 0.1‖意味着0.1是在无穷多样本的极限条件下 能够被观察到的比例，但 即使是严格的频率主义者的见解也涉及到主 观分析—reference class

例如，为什么智能体可以不支持如下的 信度集合:
P(A)=0.4, P(B)=0.3, P(AB) =0.8, and P(AB) =0.

Bruno de Finetti的论点（1931）：
如果智能体1表达了一组违反概率理论公理的 信度，那么存在一个赌局的组合保证智能体 2每次都能赢智能体1的钱。 例子：图13.2
关于不确定性的推理


必要性：处理智能体的精度和有效性的 折中 及时赶上 理性决策的含义

假设A 定所拥有的环境信息下，它是所有可以被执 行的规划中被期望能够使智能体的性能度量 达到最大的那个。
飞机、等 待时间不 90是智能体该做的事，这意味着在给 长，…
第十三章、不确定性

不确定环境下的行动 不确定性推理的方法
能进行重复试验
Make decisions under uncertainty (rep by probability)
Suppose I believe the following:
P(A25 gets me there on time | …) P(A90 gets me there on time | …) P(A120 gets me there on time | …) P(A1440 gets me there on time | …) = = = = 0.04 0.70 0.95 0.9999
• Start with the joint probability distribution: •
• Can also compute conditional probabilities: • P(cavity | toothache) = P(cavity toothache) P(toothache) = 0.016+0.064 0.108 + 0.012 + 0.016 + 0.064 = 0.4

For any propositions A, 0 ≤ P(A) P(true) = 1 For infinitely many mutual disjoint propositions A1, A2,, An,,
P( Ai ) P( Ai )
i 1 i 1


为什么概率公理是合理的？


Obvious problems:
n
第十三章、不确定性

不确定环境下的行动 不确定性推理的方法

概率



概率公理 使用全概率分布进行推理 独立性 贝叶斯法则及其应用




Fuzzy logic: an event can be ―sort of‖—vagueness, different from uncertainty.
Use interval-valued degrees of belief to represent an agent’s knowledge of the probability of a proposition Ex: coin flipping, the interval for Head is [0.1] before expert testimony and [0.45,0.55] after.
Hence a purely logical approach either
1. 2.
risks falsehood: ―A25 will get me there on time‖, or leads to conclusions that are too weak for decision making:
General idea: compute distribution on query variable by fixing evidence variables and summing over hidden variables
Inference by enumeration (contd)
Typically, we are interested in the posterior joint distribution of the query variables Y given specific values e for the evidence variables E Let the hidden variables be H = X - Y - E Then the required summation of joint entries is done by summing out the hidden variables:
Normalization
• Denominator can be viewed as a normalization constant α •
P(Cavity | toothache) = αP(Cavity,toothache)
= α[P(Cavity,toothache,catch) + P(Cavity,toothache, catch)] = α [<0.108,0.016> + <0.012,0.064>] = α<0.12,0.08> = <0.6,0.4>
P(Y | E = e) = αP(Y,E = e) = αΣhP(Y,E= e, H = h)

The terms in the summation are joint entries because Y, E and H together exhaust the set of random variables
1. 2. 2ability (road state, other drivers' plans, etc.) noisy sensors (traffic reports) uncertainty in action outcomes (flat tire, etc.) immense complexity of modeling and predicting traffic